Workshop em Fundamentos da Ciência da Computação: Algoritmos Combinatórios e Estruturas Discretas

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1 Workshop em Fundamentos da Ciência da Computação: Algoritmos Combinatórios e Estruturas Discretas IMPA, Rio de Janeiro, 1013 de abril de 2006 Programme Mo (10/4) Tu (11/4) We (12/4) Th (13/4) 9:3010:15 Füredi Duus Kostochka Mubayi Coee break 10:1510:45 10:4511:30 Moreira Szwarcter Skokan Lemos Lunch break 11:3014:00 14:0014:45 Fernandes Marciniszyn Lopes 14:4515:30 Chataigner Dellamonica ˆ Füredi: Sets of few distances in high-dimensional normed spaces ˆ Moreira: Measuring sets of innite sequences with bounded complexity ˆ Fernandes: Approximation results on rational objectives ˆ Chataigner: On balanced connected partitions of graphs ˆ Duus: Automorphisms and endomorphisms of nite partially ordered sets ˆ Szwarcter: Linear time representation and recognition of unit circulararc graphs ˆ Marciniszyn: Two results from extremal graph theory ˆ Dellamonica: Yet another result from extremal graph theory ˆ Kostochka: Domination in cubic connected graphs ˆ Skokan: Ramsey-type questions for graphs and hypergraphs ˆ Lopes: Graded forests and rational knots ˆ Mubayi: Stability in extremal set theory ˆ Lemos: Weight distribution of the bases of a graph (or a matroid) 1

2 Abstracts Sets of few distances in high-dimensional normed spaces Zoltán Füredi, U. of Illinois at Urbana-Champaign and Rényi Institute This talk is about isosceles triangles. It is known that an equidistant set in the d-dimensional Euclidean space E d can have at most d+1 elements. Let f(n, k) denote the maximum size of a set in the normed space N with at most k pairwise distinct distances. Alon and Pudlák (2003) proved that f(l d p, 1) = O(d log d) for any xed odd integer p. For general normed spaces the upper bound is f(n, k) 2 kd (Petty 1971 for k = 1 and Swaenapol 1999 for all k). The point-set {0, 1,..., k} n gives f(l d, k) (k + 1) d. Here we consider further generalizations: A set P R d in the normed space N = (R d, ) is called k-dependent if X P, X = k determines less than ( k 2) distances. For example, for k = 3, every triangle is isosceles. Let g(n d, k) denote the maximum size of such a set. The point set {1,..., ( k 2) } d shows g(l d, k) ( ) k d. 2 Our aim is to prove that there exists c dependent only on d, k such that g(n d, k) c, independently on the norm. We show, g(n d, k) ( ) 1 c3 d d log d 2 (k 1)(k2 2k + 6). Measuring sets of innite sequences with bounded complexity Carlos Gustavo Moreira, IMPA Given an innite sequence over a nite alphabet with b letters (which can be identied with a real number written in base b), its complexity function is a function p dened on the positive integers such that p(n) is the number of factors (subsequences of consecutive terms) of length n of our innite sequence. Given a function f(n), we consider the set of all innite sequences 2

3 (or real numbers) with complexity bounded by f. We give lower and upper estimates for the number of sequences of size n which appear as factors of some sequence in this set. When we consider sets of real numbers with complexity bounded by f, it is not dicult to show that they have positive Hausdor dimension if and only if f grows exponentially fast. We study this sets in general (which, in the more interesting cases, have zero Hausdor dimension), and estimate generalized fractal dimensions of them, associated to more general gauge functions than the functions x s, s > 0 (which are used to dene the Hausdor dimension). This is joint work with Christian Mauduit (IML, Marseille). Approximation results on rational objectives Cristina Gomes Fernandes, USP We address the problem of nding approximate solutions for a class of combinatorial optimization problems with rational objectives. We show that, if there is an α-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an α-approximation (1/α-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering integer programming problems and network design problems, among others. This is joint work with José R. Correa (Adolfo Ibañez, Santiago) and Yoshiko Wakabayashi (USP). On balanced connected partitions of graphs Frédéric Chataigner, USP Let (G, w) denote a pair consisting of a connected graph G and a weight function w : V (G) Z +. For X V (G), let w(x) denote the sum of the weights of the vertices in X. We consider the following problem: given a pair (G, w) and a positive integer q 2, nd a q-partition P = (V 1, V 2,..., V q ) of V (G) such that G[V i ] is connected (1 i q) and P maximizes min{w(v i ) : 1 i q}. This problem is called Max Balanced Connected Partition and is denoted by BCP. When q 2 is xed, we denote the problem by BCP q. We show some approximability and inapproximability results for both BCP and BCP 2. These results include a proof that BCP does not admit a PTAS unless P = NP and a 2-approximation algorithm for BCP 3 on 3-connected graphs. We also mention some other known algorithmic results. 3

4 This is joint work with Liliane Salgado (UFPE) and Yoshiko Wakabayashi (USP). Automorphisms and endomorphisms of nite partially ordered sets Dwight Duus, Emory University It has been known for some time that a partially ordered set of order n must have at least 2 n endomorphisms, or order preserving maps into itself. Moreover, most ordered sets have no nontrivial automorphisms. However, it is still not known if the { } End(P ) c(n) = min : with the minimum over all P of order n, Aut(P ) tends to innity with n, in complete generality, though it has been demonstrated in many situations. We examine results for the case of bipartite orders, obtained jointly with Tomasz Šuczak. Linear time representation and recognition of unit circular-arc graphs Jayme L. Szwarcter, UFRJ In a recent paper, Durán, Gravano, McConnell, Spinrad and Tucker described an algorithm of complexity O(n 2 ) for recognizing whether a graph G with n vertices and m edges is a unit circular-arc (UCA) graph. Furthermore the following open questions were posed in the above paper: (i) Is it possible to construct a UCA model for G in polynomial time? (ii) Is it possible to construct a UCA model, whose extremes of the arcs correspond to integers of polynomial size? (iii) If (ii) is true, could such a model be constructed in polynomial time? In the present paper, we describe a characterization of UCA graphs which leads to a dierent recognition algorithm and to answering these questions, in the armative. We construct a UCA model whose extremes of the arcs correspond to integers of size O(n). The proposed algorithms, for recognizing UCA graphs and constructing UCA models, have complexities O(n + m). Furthermore, the complexities reduce to O(n), if a proper circular-arc (PCA) 4

5 model of G is already given as the input, provided the extremes of the arcs are ordered. We remark that a PCA model of G can be constructed in O(n+m) time, using the algorithm by Deng, Hell and Huang. This is joint work with Min Chih Lin (Universidad de Buenos Aires). Two results from extremal graph theory Martin Marciniszyn, ETH Zürich In the rst result, we prove the existence of many complete graphs in almost all suciently dense partitions obtained by an application of Szemerédi's regularity lemma. More precisely, we consider the number of complete graphs K l on l vertices in l-partite graphs where each partition class consists of n vertices and there is an ɛ-regular graph on m edges between any two partition classes. We show that for all β > 0, at most a β m -fraction of all such graphs contain less than the expected number of copies of K l provided ɛ is suciently small, m n 2 1/(l 1), and n is suciently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Šuczak and Rödl from 1997, and it is well known that this result implies several results for random graphs. This is joint work with S. Gerke and A. Steger. In the second result, we prove a probabilistic version of the famous theorem by Erd s and Gallai from 1959 that guarantees the existence of long cycles in graphs with suciently many edges. For any constant α > 0, suppose that an adversary deletes (α o(1))pn 2 /2 edges from a random graph G(n, p). Then asymptotically almost surely there remains a cycle of length (1 α)n in the graph provided that p n 1. This is joint work with D. Dellamonica Jr, Y. Kohayakawa, and A. Steger. Yet another result from extremal graph theory Domingos Dellamonica Jr, USP The following game is proposed. Let G = G n,p be a random graph on n vertices with edge probability p n 1/2. An adversary maliciously removes up to a little less than half the edges incident to every vertex of G, thus obtaining a graph G. We prove that asymptotically almost surely G contains a Hamiltonian cycle. This is joint work with Y. Kohayakawa, M. Marciniszyn, and A. Steger. 5

6 Domination in cubic connected graphs Alexandr V. Kostochka, U. of Illinois at Urbana-Champaign In 1996, Reed proved that the domination number γ(g) of every n-vertex graph G with minimum degree at least 3 is at most 3n/8 and conjectured that if G is cubic and connected, then γ(g) n/3. Kawarabayashi, Plummer and Saito proved that the conjecture is `close to the truth' for cubic graphs of large girth. We disprove the conjecture by constructing a sequence {G k } γ(g k=1 of connected cubic graphs with lim k ) k V (G k ) 8 23 = On the other hand, we show that γ(g) 4n/11 for every n-vertex cubic connected graph G if n > 8. Some more facts on the domination number of cubic graphs will be discussed. The talk is based on joint work with B. Stodolsky. Ramsey-type questions for graphs and hypergraphs Jozef Skokan, USP Graded forests and rational knots Pedro Lopes, Instituto Superior Técnico e IMPA Rational knots are important in many respects, in particular for their connections with DNA. We recently produced a new algorithm for retrieving the determinant of any rational knot which is based on graphs. More specically, we associate a graded forest to any rational knot. This allows us to write down its determinant in terms of the numbers of half-twists in each tassle of the rational knot. In this talk we elaborate on these graded forests and rational knots. This is joint work with Louis H. Kauman. References 1. L. H. Kauman, S. Lambropoulou, On the classication of rational tangles, Adv. in Appl. Math., 33 (2004), no. 2, L. H. Kauman, P. Lopes, Color spectra and coloring conjectures, arxiv:math.gt/ , submitted 6

7 Stability in extremal set theory Dhruv Mubayi, University of Illinois at Chicago I will begin by dening the notion of stability for monotone properties of set systems. This formulation encompasses the classical denition in extremal graph theory initiated by Erd s and Simonovits in the 60s. Various stability theorems about classical intersection-type questions will be given, for example, I will show that a nontrivial intersecting family of k-sets of almost maximum size has structure close to that guaranteed by the extremal examples of the HiltonMilner theorem. Finally, I will indicate how the stability approach can be used to prove an exact result in extremal set theory. Weight distribution of the bases of a graph (or a matroid) Manoel Lemos, UFPE Given a weighted graph, let w 1, w 2,..., w n denote the increasing sequence of all possible distinct spanning tree weights. In 1992, Mayr and Plaxton proved the following conjecture proposed by Kano: every spanning tree of weight w 1 is at most k 1 edge swaps away from some spanning tree of weight w k. We will talk about the extension of this result for matroids. We will also prove that all the four conjectures due to Kano hold for matroids provided one partitions the bases of a matroid by the weight distribuition of its elements instead of their weight. 7